We study the fair division of indivisible items with subsidies among $n$ agents, where the absolute marginal valuation of each item is at most one. Under monotone valuations (where each item is a good), Brustle et al. (2020) demonstrated that a maximum subsidy of $2(n-1)$ and a total subsidy of $2(n-1)^2$ are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most $n-1$ per agent and a total subsidy of at most $n(n-1)/2$. Moreover, we present further improved bounds for monotone valuations.

翻译：我们研究在$n$个智能体之间使用补贴进行不可分物品的公平分配问题，其中每个物品的绝对边际估值不超过1。在单调估值（每个物品均为商品）条件下，Brustle等人（2020）证明了最大补贴$2(n-1)$和总补贴$2(n-1)^2$足以保证存在一种无嫉妒性分配。在本文中，我们甚至在更广泛的模型下改进了这些界限。具体而言，我们证明：给定一个EF1分配，我们可以在多项式时间内计算出一个无嫉妒分配，其每个智能体的补贴不超过$n-1$，总补贴不超过$n(n-1)/2$。此外，我们还给出了针对单调估值的进一步改进界限。